An improved upper bound for the multicolour Ramsey number of odd cycles
Maria Axenovich, Wouter Cames van Batenburg, Oliver Janzer, Lukas Michel, Mathieu Rundström
TL;DR
The paper addresses the growth of multicolour Ramsey numbers for odd cycles $C_{2\ell+1}$ by developing a local neighbourhood framework: a $k$-local-edge-colouring with controlled chromatic numbers on neighbourhoods implies a global vertex bound via a weight-induction argument. The main contribution is a new upper bound $R_k(C_{2\ell+1}) \le (4\ell - 2)^k \cdot k^{k/\ell} + 1$, which translates unconditionally into a bound of the form $R_k(C_{2\ell+1}) \le c^k \cdot k!^{1/\ell} + 1$ with $c=(4\ell - 2) e^{1/\ell}$, confirming Fox's conjecture on the exponent. A key ingredient is the lemma that bounds $n$ by $\chi^k \cdot k^{k/\ell}$ whenever each colour's neighbourhoods have chromatic number at most $\chi$, proved via a weight-based induction. The results advance the understanding of Ramsey numbers for large numbers of colours and yield refined bounds for short monochromatic odd cycles, connecting local chromatic constraints to global extremal structure.
Abstract
We show that the $k$-colour Ramsey number of an odd cycle of length $2 \ell + 1$ is at most $(4 \ell)^k \cdot k^{k/\ell}$. This proves a conjecture of Fox and is the first improvement in the exponent that goes beyond an absolute constant factor since the work of Bondy and Erdős from 1973.